Does the series $\sum_{k=1}^\infty\left(\frac{1}{k}-\frac{1}{2^k}\right)$ diverge

Question

Is there a handy way to tell if $\sum_{k=1}^\infty\left(\frac{1}{k}-\frac{1}{2^k}\right)$ diverges or not? I have a hunch that it diverges, since it looks like the sum is just $\zeta(1)-1=\infty$. But I'm not sure one can rearrange the series as $$ \sum_{k=1}^\infty\frac{1}{k}-\sum_{k=1}^\infty\frac{1}{2^k}.$$

Is that a valid move?

Answer 1

Suppose the series $\sum\limits_{k=1}^\infty\left(\frac{1}{k}-\frac{1}{2^k}\right)$ converged. The series $\sum\limits_{k=1}^\infty\frac{1}{2^k}$ converges since it is a geometric series with common ratio $\frac12$. With two convergent series, you can add them term by term to get that $$ \sum_{k=1}^\infty\frac1k\tag{1} $$ converges.

The series in $(1)$ diverges, so the series $\sum\limits_{k=1}^\infty\left(\frac{1}{k}-\frac{1}{2^k}\right)$ diverges.

Answer 2

Hint: Try a limit comparison test with the divergent harmonic series.

Answer 3

It is a valid move if at least one of the series converges. In your case one of them does.

Or else one could note that for $k \ge 2$, $\frac{1}{2^k}\le \frac{1}{2k}$. Thus if $k\ge 2$, then $\frac{1}{k}-\frac{1}{2^k}\ge \frac{1}{k}-\frac{1}{2k}=\frac{1}{2k}$.